Math, asked by shailendra85, 11 months ago

y
sin 20° sin 40° sin 60° sin 80º = =​

Answers

Answered by Anonymous
3

Step-by-step explanation:

■ we know that

●2sinAsinB = cos (A-B) - cos (A+B)

●2cosAsinB = sin (A+B) - sin(A-B)

● sin(180-A) = sin A

now ,

■ sin20°•sin40°•sin60°•sin80°

=  \frac{1}{2} [2 sin20sin40]  \frac{ \sqrt{3} }{2}  \sin(80)  \\  =  \frac{ \sqrt{3} }{4} [ \cos(20 - 40) -  \cos(20 + 40)  ] \sin(80)  \\  =  \frac{ \sqrt{3} }{4} [ \cos( - 20) -  \cos(60)  ] \sin(80)   \\ =  \frac{ \sqrt{3} }{4} [ \cos(20)  -  \frac{1}{2} ] \sin(80)  \\  =  \frac{ \sqrt{3} }{4} [ \cos(20) \sin(80)   -  \frac{1}{2}  \sin(80) ] \\  =  \frac{ \sqrt{3} }{4} [ \frac{1}{2} (2 \cos(20) \sin(80) ) -  \frac{1}{2}  \sin(80)  ] \\  =  \frac{ \sqrt{3} }{8} [2 \cos(20) \sin(80)   -  \sin(80) ] \\  =  \frac{ \sqrt{3} }{8} [ \sin(20 + 80) -   \sin(20 - 80) -  \sin(80)  ] \\  =  \frac{ \sqrt{3} }{8} [ \sin(100)  -  \sin( - 60)  -  \sin(80) ] \\  =  \frac{ \sqrt{3} }{8} [ \sin(180 - 80) -  \sin( - 60)  -  \sin(80)  ] \\  =  \frac{ \sqrt{3} }{8} [ \sin(80)  -  \sin( - 60) -  \sin(80)  ] \\  =  \frac{ \sqrt{3} }{8} [ -  \sin( - 60) ] \\  =  \frac{ \sqrt{3} }{8} [ \sin(60) ] \\  =  \frac{ \sqrt{3} }{8} [ \frac{ \sqrt{3} }{2} ] \\  =  \frac{3}{16}

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